"As President Trump has stated, consistent with USA law, the United States will not allow illegal immigrants to enter or remain in the United States", Pompeo stated.
A ragged army of Honduran migrants streamed through southern Mexico on Sunday heading toward the United States, after making an end-run around Mexican agents who briefly blocked them at the Guatemalan border.
The migrants have defied threats by Mr Trump that he will close the US-Mexico border if the caravan advances and warnings from the Mexican Government that they risk deportation if they can not justify seeking asylum in Mexico.
Lopez Obrador, who takes office December 1, said people who leave their home do so not because they want to, but out of necessity.
It was not immediately clear where the additional travellers had materialized from since about 2,000 gathered on the Mexican side Saturday night.
Jesus Valdivia, of Tuxtla Chico, Mexico, was one of the many who pulled his pickup truck over to let 10 or even 20 migrants hop in at a time, sometimes causing vehicles' springs to groan under the weight.
The original caravan left San Pedro Sula, Honduras, one of the country's most violent cities, on October 13 which quickly accumulated to include a now reported 8,000 people who are dispersed across Honduras, Guatemala and now Mexico.
"The Mexican government should put the human rights of the caravan members at the heart of the response and not let the U.S. Trump administration's pressure prevent it from fulfilling its global obligations", said Erika Guevara-Rosas, the organization's director for the region.
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Mexico's National Migration Institute said it reiterates its duty to safeguard the human rights of migrants who enter its territory.
The multitude of migrants clogged the highway leading north from the border city of Ciudad Hidalgo in the southern Mexican state of Chiapas, many breaking into song.
"They catch you, and you try to get back", said Imner Anthony Fuentes, a 29-year-old who reportedly was deported for the sixth time from the US five months ago.
Osman Melgar, who nursed a bleeding gash on his shin, suffered when he fell as dozens of people packed on the bridge began fleeing Mexican police using tear gas, according to several eyewitnesses.
Last week, Trump threatened to cut aid to the region, deploy the military, and close the US-Mexican border if authorities did not stop them.
Mexico's interior department said it had received 640 refugee requests by Hondurans at the border crossing.
"This is an exodus", said Ruben Figuerora of the Mesoamerican Migrant Movement, standing on the muddy bank of the Mexican side of the Suchite river, as a line of young male migrants walked past him after crossing on a raft. The advances have overwhelmed Mexican officials' attempts to stop them at the border.
The decision to re-form the migrant caravan capped a day in which Mexican authorities again refused mass entry to migrants on the bridge, instead accepting small groups for asylum processing and giving out 45-day visitor permits to some. But the 25-year-old farmer was determined to press onward, explaining that the $6 a day he made back home was not enough to live on.
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"Full efforts are being made to stop the onslaught of illegal aliens from crossing our [southern] Border", Trump tweeted Sunday.
He took to Twitter to blame the courts for "asking the United States to do things that are not doable".
"I'm looking for a better future for my children".
US Department of Homeland Security Secretary Kirstjen Nielsen noted that her agency is monitoring the caravan while going so far as to suggest that the migrant movement can serve as a cover for transnational criminal organizations.
Human rights groups have criticised the U.S. and Mexican response to the caravan.
Presidents Hernandez of Honduras and Jimmy Morales of Guatemala held an emergency meeting Saturday at a Guatemalan.
Migrants have commonly cited widespread poverty and gang violence in Honduras, one of the world's deadliest nations by homicide rate, as their reasons for joining the caravan.
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